A326361 -id:A326361 - cp's OEIS frontend

A001206 Number of self-dual monotone Boolean functions of n variables.

0, 1, 2, 4, 12, 81, 2646, 1422564, 229809982112, 423295099074735261880

Offset: 0

Sometimes called Hosten-Morris numbers (or HM numbers).
Also the number of simplicial complexes on the set {1, ..., n-1} such that no pair of faces covers all of {1, ..., n-1}. [Miller-Sturmfels] - N. J. A. Sloane, Feb 18 2008
Also the maximal number of generators of a neighborly monomial ideal in n variables. [Miller-Sturmfels]. - N. J. A. Sloane, Feb 18 2008
Also the number of intersecting antichains on a labeled (n-1)-set or (n-1)-variable Boolean functions in the Post class F(7,2). Cf. A059090. - Vladeta Jovovic, Goran Kilibarda, Dec 28 2000
Also the number of nondominated coteries on n members. - Don Knuth, Sep 01 2005
The number of maximal families of intersecting subsets of an n-element set. - Bridget Tenner, Nov 16 2006
Rivière gives a(n) for n <= 5. - N. J. A. Sloane, May 12 2012

			a(2) = 1 + 1 = 2;
a(3) = 1 + 3 = 4;
a(4) = 1 + 7 + 3 + 1 = 12;
a(5) = 1 + 15 + 30 + 30 + 5 = 81;
a(6) = 1 + 31 + 195 + 605 + 780 + 543 + 300 + 135 + 45 + 10 + 1 = 2646;
a(7) = 1 + 63 + 1050 + 9030 + 41545 + 118629 + 233821 + 329205 + 327915 + 224280 + 100716 + 29337 + 5950 + 910 + 105 + 1 = 1422564.
Cf. A059090.
From _Gus Wiseman_, Jul 03 2019: (Start)
The a(1) = 1 through a(4) = 12 intersecting antichains of nonempty sets (see Jovovic and Kilibarda's comment):
  {}  {}     {}       {}
      {{1}}  {{1}}    {{1}}
             {{2}}    {{2}}
             {{1,2}}  {{3}}
                      {{1,2}}
                      {{1,3}}
                      {{2,3}}
                      {{1,2,3}}
                      {{1,2},{1,3}}
                      {{1,2},{2,3}}
                      {{1,3},{2,3}}
                      {{1,2},{1,3},{2,3}}
(End)

Martin Aigner and Günter M. Ziegler, Proofs from THE BOOK, Third Edition, Springer-Verlag, 2004. See chapter 22.
V. Jovovic and G. Kilibarda, The number of n-variable Boolean functions in the Post class F(7,2), Belgrade, 2001, in preparation.
D. E. Knuth, The Art of Computer Programming, Vol. 4A, Section 7.1.1, p. 79.
W. F. Lunnon, The IU function: the size of a free distributive lattice, pp. 173-181 of D. J. A. Welsh, editor, Combinatorial Mathematics and Its Applications. Academic Press, NY, 1971.
Charles F. Mills and W. M. Mills, The calculation of λ(8), preprint, 1979. Gives a(8).
E. Miller and B. Sturmfels, Combinatorial Commutative Algebra, Springer, 2005.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Taras Banakh, Volodymyr Gavrylkiv, Automorphism groups of superextensions of groups, arXiv:1802.05804 [math.GR], 2018.
Jan C. Bioch and Toshihide Ibaraki, Generating and approximating nondominated coteries, IEEE Transactions on parallel and distributed systems 6 (1995), 905-914.
A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Report ZN 41, March 1972, Math. Centr., Amsterdam. Gives a(n) for n <= 7.
A. E. Brouwer and A. Verbeek, Counting families of mutually intersecting sets, Electronic Journal of Combinatorics, Volume 20, Issue 2 (2013), Paper #P8.
Gábor Damásdi, Stefan Felsner, António Girão, Balázs Keszegh, David Lewis, Dániel T. Nagy, Torsten Ueckerdt, On Covering Numbers, Young Diagrams, and the Local Dimension of Posets, arXiv:2001.06367 [math.CO], 2020.
Jesús A. De Loera, Serkan Hoşten, Robert Krone, Lily Silverstein, Average Behavior of Minimal Free Resolutions of Monomial Ideals, arXiv:1802.06537 [math.AC], 2018.
Serkan Hosten and Walter D. Morris, Jr., The order dimension of the complete graph, Discrete Math. 201 (1999), pp. 133-139.
D. E. Loeb, Challenges in playing multiplayer games, in Levy and Beal, editors, Heuristic Programming in Artificial Intelligence, vol. 4, Ellis Horwood, 1994. [broken link]
D. E. Loeb and A. Meyerowitz, The maximal intersecting family of sets graph, in H. Barcelo and G. Kalai, editors, Proceedings of the Conference on Jerusalem Combinatorics 1993. AMS series Contemporary Mathematics, 1994.
Bartłomiej Pawelski, Counting and generating monotone Boolean functions, Doctoral Diss., Univ. Gdańsk, (Poland, 2024). See pp. 12, 15, 38, 49, 58, 73.
Bartlomiej Pawelski and Andrzej Szepietowski, Divisibility properties of Dedekind numbers, arXiv:2302.04615 [math.CO], 2023.
N. M. Rivière, Recursive formulas on free distributive lattices, J. Combinatorial Theory 5 1968 229--234. MR0231764 (38 #92).
Tom Trotter, An Application of the Erdős/Stone Theorem, Slides, Sept. 13, 2001.
Index entries for sequences related to Boolean functions

Cf. A000372, A059090.
The case with empty edges allowed is A326372.
The maximal case is A007363, or A326363 with empty edges allowed.
The case with empty intersection is A326366.
The inverse binomial transform is the covering case A305844.
Cf. A006126, A014466, A051185, A326361, A326362.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]],{n,0,5}] (* Gus Wiseman, Jul 03 2019 *)

a(n+1) = Sum_{m=0..A037952(n)} A059090(n, m).

For n > 0, a(n) = A326372(n - 1) - 1. - Gus Wiseman, Jul 03 2019

a(8) due to C. F. Mills & W. H. Mills, 1979
a(8) from Daniel E. Loeb, Jan 04 1996
a(8) confirmed by Don Knuth, Feb 08 2008
a(9) from Andries E. Brouwer, Aug 25 2012

A326358 Number of maximal antichains of subsets of {1..n}.

Original entry on oeis.org

1, 2, 3, 7, 29, 376, 31746, 123805914

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(0) = 1 through a(3) = 7 maximal antichains:
  {}  {}   {}      {}
      {1}  {12}    {123}
           {1}{2}  {1}{23}
                   {2}{13}
                   {3}{12}
                   {1}{2}{3}
                   {12}{13}{23}

Denis Bouyssou, Thierry Marchant, and Marc Pirlot, ELECTRE TRI-nB, pseudo-disjunctive: axiomatic and combinatorial results, arXiv:2410.18443 [cs.DM], 2024. See p. 14.
Dmitry I. Ignatov, A Note on the Number of (Maximal) Antichains in the Lattice of Set Partitions. In: Ojeda-Aciego, M., Sauerwald, K., Jäschke, R. (eds) Graph-Based Representation and Reasoning. ICCS 2023. Lecture Notes in Computer Science(). Springer, Cham.
Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.

Antichains of sets are A000372.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305001, A305844, A326360, A326361, A326362, A326363.

LoadPackage("grape");
      maxachP:=function(n) local g,G;
       g:=Graph(Group(()), Combinations([1..n]), function(x, g) return x; end,
          function(x, y) return not IsSubset(x, y) and not IsSubset(y, x); end, true);
       G:=AutGroupGraph(g);
       return Sum(CompleteSubgraphs(NewGroupGraph(G, g), -1, 2),
              function(c) return Length(Orbit(G, c, OnSets)); end);
     end;
       List([0..7],maxachP); # Mamuka Jibladze, Jan 26 2021

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n]],SubsetQ]]],{n,0,5}]
(* alternatively *)
maxachP[n_]:=FindIndependentVertexSet[
  Flatten[Map[Function[s, Map[# \[DirectedEdge] s &, Most[Subsets[s]]]],
    Subsets[Range[n]]]], Infinity, All];
Table[Length[maxachP[n]],{n,0,6}] (* Mamuka Jibladze, Jan 25 2021 *)

For n > 0, a(n) = A326359(n) + 1.

a(6)-a(7) from Mamuka Jibladze, Jan 26 2021

A007363 Maximal self-dual antichains on n points.

Original entry on oeis.org

0, 1, 3, 5, 20, 168, 11748, 12160647

Offset: 1

N. J. A. Sloane

From Gus Wiseman, Jul 02 2019: (Start)
If self-dual means (pairwise) intersecting, then a(n) is the number of maximal intersecting antichains of nonempty subsets of {1..(n - 1)}. A set of sets is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint. For example, the a(2) = 1 through a(5) = 20 maximal intersecting antichains are:
   {1}    {1}     {1}             {1}
          {2}     {2}             {2}
          {12}    {3}             {3}
                  {123}           {4}
                  {12}{13}{23}    {1234}
                                  {12}{13}{23}
                                  {12}{14}{24}
                                  {13}{14}{34}
                                  {23}{24}{34}
                                  {12}{134}{234}
                                  {13}{124}{234}
                                  {14}{123}{234}
                                  {23}{124}{134}
                                  {24}{123}{134}
                                  {34}{123}{124}
                                  {12}{13}{14}{234}
                                  {12}{23}{24}{134}
                                  {13}{23}{34}{124}
                                  {14}{24}{34}{123}
                                  {123}{124}{134}{234}
(End)

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Daniel E. Loeb, On Games, Voting Schemes and Distributive Lattices. LaBRI Report 625-93, University of Bordeaux I, 1993. [Broken link]

Intersecting antichains are A326372.
Intersecting antichains of nonempty sets are A001206.
Unlabeled intersecting antichains are A305857.
Maximal antichains of nonempty sets are A326359.
The case with empty edges allowed is A326363.
Cf. A000372, A305844, A306007, A326358, A326361, A326362, A326366.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}] (* Gus Wiseman, Jul 02 2019 *)
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 0, a(n) = A326363(n - 1) - 1 = A326362(n - 1) + n - 1. - Gus Wiseman, Jul 03 2019

a(8) from Elijah Beregovsky, May 06 2020

A326363 Number of maximal intersecting antichains of subsets of {1..n}.

Original entry on oeis.org

1, 2, 4, 6, 21, 169, 11749, 12160648

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other, and is intersecting if no two element are disjoint.

			The a(1) = 1 through a(4) = 21 maximal intersecting antichains:
  {}   {}    {}            {}
  {1}  {1}   {1}           {1}
       {2}   {2}           {2}
       {12}  {3}           {3}
             {123}         {4}
             {12}{13}{23}  {1234}
                           {12}{13}{23}
                           {12}{14}{24}
                           {13}{14}{34}
                           {23}{24}{34}
                           {12}{134}{234}
                           {13}{124}{234}
                           {14}{123}{234}
                           {23}{124}{134}
                           {24}{123}{134}
                           {34}{123}{124}
                           {12}{13}{14}{234}
                           {12}{23}{24}{134}
                           {13}{23}{34}{124}
                           {14}{24}{34}{123}
                           {123}{124}{134}{234}

The case with nonempty, non-singleton edges is A326362.
Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{0,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets] (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A007363(n + 1) + 1 = A326362(n) + n + 1.

a(7) from Elijah Beregovsky, May 06 2020

A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 16, 163, 11742, 12160640

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 16 maximal intersecting antichains:
  {{1,2,3,4}}
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,4},{2,4}}
  {{1,3},{1,4},{3,4}}
  {{2,3},{2,4},{3,4}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A051185, A058891, A261005, A305000, A305844, A326358, A326360, A326361, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&]]],{n,0,5}]
(* 2nd program *)
n = 2^6; g = CompleteGraph[n]; i = 0;
While[i < n, i++; j = i; While[j < n, j++; If[BitAnd[i, j] == 0 || BitAnd[i, j] == i || BitAnd[i, j] == j, g = EdgeDelete[g, i <-> j]]]];
sets = FindClique[g, Infinity, All];
Length[sets]-Log[2,n]-1 (* Elijah Beregovsky, May 06 2020 *)

For n > 1, a(n) = A326363(n) - n - 1 = A007363(n + 1) - n.

a(7) from Elijah Beregovsky, May 06 2020

A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

Original entry on oeis.org

1, 1, 1, 2, 13, 279, 29820, 123590767

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no element is a subset of any other.

			The a(1) = 1 through a(4) = 13 maximal antichains:
  {}  {12}  {123}         {1234}
            {12}{13}{23}  {12}{134}{234}
                          {13}{124}{234}
                          {14}{123}{234}
                          {23}{124}{134}
                          {24}{123}{134}
                          {34}{123}{124}
                          {12}{13}{14}{234}
                          {12}{23}{24}{134}
                          {13}{23}{34}{124}
                          {14}{24}{34}{123}
                          {123}{124}{134}{234}
                          {12}{13}{14}{23}{24}{34}

Dmitry I. Ignatov, On the Number of Maximal Antichains in Boolean Lattices for n up to 7. Lobachevskii J. Math., 44 (2023), 137-146.
Dmitry I. Ignatov, Supporting iPython code and input files for a(7) based on inequivalent maximal antichains for n=7 and related sequences, Github repository, section 3
Dmitry I. Ignatov, PDF version of the supporting iPython notebook for a(7)
Dmitry I. Ignatov, Supporting iPython notebook for a(7): A326360.ipynb

Antichains of nonempty, non-singleton sets are A307249.
Minimal covering antichains are A046165.
Maximal intersecting antichains are A007363.
Maximal antichains of nonempty sets are A326359.
Cf. A000372, A003182, A006126, A006602, A014466, A058891, A261005, A305000, A305844, A326358, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[stableSets[Subsets[Range[n],{2,n}],SubsetQ]]],{n,0,4}]

# see Ignatov links
# Dmitry I. Ignatov, Oct 14 2021

a(n) = Sum_{k=0..n} (-1)^(n-k)*binomial(n,k)*A326359(k) for n >= 2. - Andrew Howroyd, Nov 19 2021

a(6) from Andrew Howroyd, Aug 14 2019

a(7) from Dmitry I. Ignatov, Oct 14 2021

A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 1, 23, 1834, 1367903, 229745722873, 423295077919493525420

Offset: 0

Gus Wiseman, Jul 01 2019

Covering means there are no isolated vertices. A set system (set of sets) is an antichain if no part is a subset of any other, and is intersecting if no two parts are disjoint.

			The a(4) = 23 intersecting antichains with empty intersection:
  {{1,2},{1,3},{2,3,4}}
  {{1,2},{1,4},{2,3,4}}
  {{1,2},{2,3},{1,3,4}}
  {{1,2},{2,4},{1,3,4}}
  {{1,3},{1,4},{2,3,4}}
  {{1,3},{2,3},{1,2,4}}
  {{1,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{1,2,3}}
  {{1,4},{3,4},{1,2,3}}
  {{2,3},{2,4},{1,3,4}}
  {{2,3},{3,4},{1,2,4}}
  {{2,4},{3,4},{1,2,3}}
  {{1,2},{1,3,4},{2,3,4}}
  {{1,3},{1,2,4},{2,3,4}}
  {{1,4},{1,2,3},{2,3,4}}
  {{2,3},{1,2,4},{1,3,4}}
  {{2,4},{1,2,3},{1,3,4}}
  {{3,4},{1,2,3},{1,2,4}}
  {{1,2},{1,3},{1,4},{2,3,4}}
  {{1,2},{2,3},{2,4},{1,3,4}}
  {{1,3},{2,3},{3,4},{1,2,4}}
  {{1,4},{2,4},{3,4},{1,2,3}}
  {{1,2,3},{1,2,4},{1,3,4},{2,3,4}}

Intersecting antichain covers are A305844.
Intersecting covers with empty intersection are A326364.
Antichain covers with empty intersection are A305001.
The binomial transform is the non-covering case A326366.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305843, A307249, A318128, A318129, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326366 Number of intersecting antichains of nonempty subsets of {1..n} with empty intersection (meaning there is no vertex in common to all the edges).

Original entry on oeis.org

1, 1, 1, 2, 28, 1960, 1379273, 229755337549, 423295079757497714059

Offset: 0

Gus Wiseman, Jul 01 2019

A set system (set of sets) is an antichain if no edge is a subset of any other, and is intersecting if no two edges are disjoint.

			The a(0) = 1 through a(4) = 28 intersecting antichains with empty intersection:
  {}  {}  {}  {}              {}
              {{12}{13}{23}}  {{12}{13}{23}}
                              {{12}{14}{24}}
                              {{13}{14}{34}}
                              {{23}{24}{34}}
                              {{12}{13}{234}}
                              {{12}{14}{234}}
                              {{12}{23}{134}}
                              {{12}{24}{134}}
                              {{13}{14}{234}}
                              {{13}{23}{124}}
                              {{13}{34}{124}}
                              {{14}{24}{123}}
                              {{14}{34}{123}}
                              {{23}{24}{134}}
                              {{23}{34}{124}}
                              {{24}{34}{123}}
                              {{12}{134}{234}}
                              {{13}{124}{234}}
                              {{14}{123}{234}}
                              {{23}{124}{134}}
                              {{24}{123}{134}}
                              {{34}{123}{124}}
                              {{12}{13}{14}{234}}
                              {{12}{23}{24}{134}}
                              {{13}{23}{34}{124}}
                              {{14}{24}{34}{123}}
                              {{123}{124}{134}{234}}

The case with empty edges allowed is A326375.
Intersecting antichains of nonempty sets are A001206.
Intersecting set systems with empty intersection are A326373.
Antichains of nonempty sets with empty intersection are A006126 or A307249.
The inverse binomial transform is the covering case A326365.
Cf. A007363, A014466, A051185, A058891, A305001, A305843, A305844, A318128, A318129, A326361, A326362, A326363, A326364.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Or[Intersection[#1,#2]=={},SubsetQ[#1,#2]]&],#=={}||Intersection@@#=={}&]],{n,0,4}]

a(n) = A326375(n) - 1.

a(n) = A001206(n+1) + A307249(n) - A014466(n). - Andrew Howroyd, Aug 14 2019

a(7)-a(8) from Andrew Howroyd, Aug 14 2019

A326364 Number of intersecting set systems with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

Original entry on oeis.org

1, 0, 0, 2, 426, 987404, 887044205940, 291072121051815578010398, 14704019422368226413234332571239460300433492086, 12553242487939461785560846872353486129110194397301168776798213375239447299205732561174066488

Offset: 0

Gus Wiseman, Jul 01 2019

nonn

Covering means there are no isolated vertices. A set system (set of sets) is intersecting if no two edges are disjoint.

			The a(3) = 2 intersecting set systems with empty intersection:
  {{1,2},{1,3},{2,3}}
  {{1,2},{1,3},{2,3},{1,2,3}}

Covering set systems with empty intersection are A318128.
Covering, intersecting set systems are A305843.
Covering, intersecting antichains with empty intersection are A326365.
Cf. A006126, A007363, A014466, A051185, A058891, A305844, A307249, A318129, A326361, A326362, A326363.

stableSets[u_,Q_]:=If[Length[u]==0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r==w||Q[r,w]||Q[w,r]],Q]]]];
Table[Length[Select[stableSets[Subsets[Range[n],{1,n}],Intersection[#1,#2]=={}&],And[Union@@#==Range[n],#=={}||Intersection@@#=={}]&]],{n,0,4}]

Inverse binomial transform of A326373. - Andrew Howroyd, Aug 12 2019

a(6)-a(9) from Andrew Howroyd, Aug 12 2019

A329561 BII-numbers of intersecting antichains of sets.

Original entry on oeis.org

0, 1, 2, 4, 8, 16, 20, 32, 36, 48, 52, 64, 128, 256, 260, 272, 276, 320, 512, 516, 544, 548, 576, 768, 772, 832, 1024, 1040, 1056, 1072, 1088, 2048, 2064, 2080, 2096, 2112, 2304, 2320, 2368, 2560, 2592, 2624, 2816, 2880, 3072, 3088, 3104, 3120, 3136, 4096

Offset: 1

Gus Wiseman, Nov 28 2019

nonn

A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system (finite set of finite nonempty sets of positive integers) has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.

A set-system is intersecting if no two edges are disjoint. It is an antichain if no edge is a proper subset of any other.

			The sequence of terms together with their corresponding set-systems begins:
    0: {}
    1: {{1}}
    2: {{2}}
    4: {{1,2}}
    8: {{3}}
   16: {{1,3}}
   20: {{1,2},{1,3}}
   32: {{2,3}}
   36: {{1,2},{2,3}}
   48: {{1,3},{2,3}}
   52: {{1,2},{1,3},{2,3}}
   64: {{1,2,3}}
  128: {{4}}
  256: {{1,4}}
  260: {{1,2},{1,4}}
  272: {{1,3},{1,4}}
  276: {{1,2},{1,3},{1,4}}
  320: {{1,2,3},{1,4}}
  512: {{2,4}}
  516: {{1,2},{2,4}}

Intersection of A326704 (antichains) and A326910 (intersecting).
Covering intersecting antichains of sets are counted by A305844.
BII-numbers of antichains with empty intersection are A329560.
Cf. A000120, A048143, A048793, A070939, A087086, A305857, A306007, A326031, A326361, A326912, A329628.

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Select[Range[0,1000],stableQ[bpe/@bpe[#],SubsetQ[#1,#2]||Intersection[#1,#2]=={}&]&]

cp's OEIS Frontend

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A326358 Number of maximal antichains of subsets of {1..n}.

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A007363 Maximal self-dual antichains on n points.

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A326363 Number of maximal intersecting antichains of subsets of {1..n}.

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A326362 Number of maximal intersecting antichains of nonempty, non-singleton subsets of {1..n}.

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A326360 Number of maximal antichains of nonempty, non-singleton subsets of {1..n}.

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A326365 Number of intersecting antichains with empty intersection (meaning there is no vertex in common to all the edges) covering n vertices.

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